To solve the given system of linear equations:
[tex]\[
\begin{cases}
3x - y + z = 7 \\
2x + y - 2z = 5 \\
4x + 7y + 5z = 1
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations simultaneously.
### Step 1: Write the equations in matrix form
We can represent the system of equations as a matrix equation [tex]\( A \mathbf{x} = \mathbf{b} \)[/tex] where
[tex]\[
A = \begin{pmatrix}
3 & -1 & 1 \\
2 & 1 & -2 \\
4 & 7 & 5
\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix}
7 \\
5 \\
1
\end{pmatrix}
\][/tex]
### Step 2: Use an appropriate method to solve the linear system
Without carrying out the detailed calculations here, we apply methods such as substitution, elimination, or matrix operations (like Gaussian elimination or using the inverse of matrix [tex]\( A \)[/tex]) to solve the system.
### Step 3: Find the solution
The result from solving this system is:
[tex]\[
x = \frac{194}{85}, \quad y = -\frac{63}{85}, \quad z = -\frac{10}{17}
\][/tex]
### Step 4: Interpret the solution
Therefore, the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations are:
[tex]\[
x = \frac{194}{85}, \quad y = -\frac{63}{85}, \quad z = -\frac{10}{17}
\][/tex]
These result in the exact solution to the given system of linear equations.
